biogeography-based optimization for different economic load dispatch problems

1064 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 2, MAY 2010

Biogeography-Based Optimization for DifferentEconomic Load Dispatch ProblemsAniruddha Bhattacharya, Member, IEEE, and Pranab Kumar Chattopadhyay

Abstract—This paper presents a biogeography-based optimiza-tion (BBO) algorithm to solve both convex and non-convex eco-nomic load dispatch (ELD) problems of thermal plants. The pro-posed methodology can take care of economic dispatch problemsinvolving constraints such as transmission losses, ramp rate limits,valve point loading, multi-fuel options and prohibited operatingzones. Biogeography deals with the geographical distribution ofbiological species. Mathematical models of biogeography describehow a species arises, migrates from one habitat to another and getswiped out. BBO has some features that are in common with otherbiology-based optimization methods, like genetic algorithms (GAs)and particle swarm optimization (PSO). This algorithm searchesfor the global optimum mainly through two steps: migration andmutation. The effectiveness of the proposed algorithm has beenverified on four different test systems, both small and large, in-volving varying degree of complexity. Compared with the otherexisting techniques, the proposed algorithm has been found to per-form better in a number of cases. Considering the quality of thesolution obtained, this method seems to be a promising alternativeapproach for solving the ELD problems in practical power system.

Index Terms—Biogeography-based optimization, economic loaddispatch, genetic algorithm, particle swarm optimization, prohib-ited operating zone.

I. INTRODUCTION

E CONOMIC load dispatch (ELD) seeks “the best” gen-eration schedule for the generating plants to supply the

required demand plus transmission losses at minimum produc-tion cost. Various investigations on ELD have been undertakenuntil date, as better solutions would result in significant eco-nomical benefits. Previously a number of derivative-based ap-proaches including Lagrangian multiplier method [1] have beenapplied to solve ELD problems. These methods require that in-cremental cost curves are monotonically increasing in nature.But in practice, the input output characteristics of modern gen-erating units are highly nonlinear due to valve-point loadings,ramp-rate limits, multi-fuel options, etc. Their characteristicshave to be approximated to meet the requirements of classicaldispatch algorithms. Due to such approximation the solutionis only sub-optimal and hence a huge amount of revenue loss

Manuscript received February 03, 2009; revised June 05, 2009. First pub-lished December 01, 2009; current version published April 21, 2010. This workwas supported by the Electrical Engineering Department, Jadavpur University.Paper no. TPWRS-00077-2009.

The authors are with Jadavpur University, Kolkata, West Bengal 700 032,India (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRS.2009.2034525

occurs over the time. Highly nonlinear characteristics of theseunits demand for solution techniques that have no restrictions onto the shape of the fuel cost curves. The calculus-based methodsfail in solving these types of problems. Wood and Wollenbergproposed dynamic programming [2], which does not imposeany restriction on the nature of the cost curves and solves bothconvex and non-convex ELD problems. But this method suffersfrom the curse of dimensionality and simulation time increasesrapidly with the increase of system size. More interests havebeen focused on the application of artificial intelligence tech-nology for solution of ELD problems. Several methods, such asgenetic algorithm (GA) [3]; artificial neural networks [4]; sim-ulated annealing (SA), Tabu search; evolutionary programming[5]; particle swarm optimization (PSO) [6]; ant colony optimiza-tion; differential evolution [7]; etc. have been developed and ap-plied successfully to ELD problems.

Yalcinoz and Short [8] have also implemented Hopfieldneural networks to solve ELD problems for units with piece-wise quadratic fuel cost functions and prohibited zonesconstraint. However, due to adoption of sigmoid function inthe model, it suffers from very slow convergence. GA and SAhave also been successfully employed to solve ELD problems.The SA method is usually slower than the GA method becausethe GA has parallel search capabilities. However, research hasidentified some deficiencies in application to highly epistaticobjective functions where the parameters being optimizedare strongly correlated. In such systems, the chromosomes inthe population, towards the end of the evolutionary processhave similar structures and their average fitness is high. Thecrossover and mutation operations cannot ensure better fitnessof offspring and therefore degradation in efficiency is apparent.Moreover, due to the premature convergence of GA, its perfor-mance degrades and its search capability reduces.

In the mid 1990s, Kennedy and Eberhart invented PSO [6]. InPSO there are only a few parameters to be adjusted, which makePSO more attractive. Simple concept, easy implementation, ro-bustness and computational efficiency are the main advantagesof the PSO algorithm. A closer examination on the operationof the algorithm indicates that once inside the optimum region,the algorithm progresses slowly due to its inability to adjust thevelocity step size to continue the search at a finer grain. So formulti-modal function, particles sometimes fail to reach globaloptimal point.

Price and Storn invented differential evolution (DE) [7]. It in-volves three basic operations, e.g., mutation, crossover, and se-lection, in order to reach an optimal solution. DE has been foundto yield better and faster solution, satisfying all the constraints,both for uni-modal and multi-modal system, using its different

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BHATTACHARYA AND CHATTOPADHYAY: BIOGEOGRAPHY-BASED OPTIMIZATION 1065

crossover strategies. But when system complexity and size in-creases, DE method is unable to map its entire unknown vari-ables together in a better way. In DE all variables are changedtogether during the crossover operation. The individual variableis not tuned separately. So in starting stage, the solutions movesvery fast towards the optimal point but at later stage when finetuning operation is required, DE fails to give better performance.

Artificial Immune System (AIS) [9] is another population-based or network-based soft computing technique in the field ofoptimization that has been successfully implemented in variouspower system optimization problems. In each iteration of AIS,many operations like affinity calculation, cloning, hyper-muta-tion, and selection are performed. During cloning, operation sizeof population also increases. Due to increase in number of oper-ations, and larger size of population, convergence speed of AISis much slower than DE or PSO.

Inspired from the mechanism of the survival of bacteria, e.g.,E. coli, an optimization algorithm, called Bacterial Foraging Al-gorithm (BFA) [10], has been developed. Chemotaxis, repro-duction and dispersion are the three processes with the help ofwhich global searching capability of this algorithm has beenachieved. These properties have helped BFA to be applied suc-cessfully in several kinds of power system optimization prob-lems. But constraints satisfaction creates little trouble in BFA.

Very recently, a new optimization concept, based on biogeog-raphy, has been proposed by Simon [11]. Biogeography is thenature’s way of distributing species. Let us consider an opti-mization problem with some trial solutions of it. In BBO, a goodsolution is analogous to an island with a high Habitat SuitabilityIndex (HSI), and a poor solution represents an island with a lowHSI. High HSI solutions resist change more than low HSI solu-tions. Low HSI solutions tend to copy good features from highHSI solutions. The shared features remain in the high HSI so-lutions, while at the same time appearing as new features in thelow HSI solutions. This is as if some representatives of a speciesmigrating to a habitat, while other representatives remain intheir original habitat. Poor solutions accept a lot of new fea-tures from good solutions. This addition of new features to lowHSI solutions may raise the quality of those solutions. This newapproach to solve a problem is known as biogeography-basedoptimization (BBO) [11].

BBO works based on the two mechanisms: migration andmutation. BBO, as in other biology-based algorithms like GAand PSO, has the property of sharing information between solu-tions. Besides, the algorithm has certain unique features whichovercome several demerits of the conventional methods as men-tioned below.

1) In BBO and PSO, the solutions survive forever althoughtheir characteristics change as the optimization processprogresses. But solutions of evolutionary-based algo-rithms like GA, EP, DE, etc. “die” at the end of each gen-eration. Due to presence of crossover operation in evo-lutionary-based algorithms, many solutions whose fit-ness are initially good sometimes lose their quality inlater stage of the process. In BBO there is no crossover-like operation; solutions get fine tuned gradually as the

process goes on through migration operation. Elitismoperation has made the algorithm more efficient in thisrespect. This gives an edge to BBO over techniques men-tioned above.

2) In PSO, solutions are more likely to clump together insimilar groups, while in BBO, solutions do not have thetendency to cluster due to its new type of mutation oper-ation. This is an added advantage of BBO in comparisonto PSO.

3) BBO involves fewer computational steps per iterationcompared to AIS. This results in faster convergence.

4) In BBO poor solutions accept a lot of new features fromgood ones which may improve the quality of those so-lutions. This is a unique feature of BBO algorithm com-pared to other techniques. At the same time this makesconstraint satisfaction much easier, compared to that inBFA.

These versatile properties of this new algorithm encouraged theauthors to apply this newly developed algorithm to solve non-convex complex ELD problems. This paper considers four typesof non-convex ELD problems, namely, ELD with quadratic costfunction and transmission loss (ELDQCTL), ELD with prohib-ited operating zones and ramp rate limits (ELDPOZRR), ELDwith valve-point loading effects (ELDVPL), and ELD with com-bined valve-point loading effects and multi-fuel options (ELD-VPLMF).

The performance of the proposed method in terms of solutionquality and computational efficiency has been compared 1) withSOH_PSO [12] and other methods for a 40-generator systemincluding valve-point loading; 2) with Hopfield model [4] andother techniques for a 20-generator system with quadratic costfunction; 3) with NPSO_LRS [13] and other techniques for aten-generator system with multi-fuel options; and 4) with newcoding-based modified PSO [14] and other previously devel-oped techniques for a six-generator system including ramp ratelimit and prohibited operating zone.

Section II of the paper provides a brief description and math-ematical formulation of different types of ELD problems. Theconcept of biogeography is discussed in Section III. The orig-inal BBO approach is described in Section IV along with a shortdescription of the algorithm used in this test system. The param-eter settings for the test system to evaluate the performance ofBBO and the simulation studies are discussed in Section V. Theconclusion is drawn in Section VI.

II. ECONOMIC LOAD DISPATCH PROBLEMS

The ELD may be formulated as a nonlinear constrainedproblem. Both convex and non-convex ELD problems havebeen modeled in this paper. The convex ELD problem assumesquadratic cost function along with system power demand andoperational limit constraints. The practical non-convex ELD(NCELD) problem, in addition, considers generator nonlinear-ities such as valve point loading effects, prohibited operatingzones, ramp rate limits, and multi-fuel options.


A. ELDQCTL

The objective function of ELD problem may be written as

(1)

where is the th generator’s cost function, and is usuallyexpressed as a quadratic polynomial; , , and are the costcoefficients of the th generator; is the number of committedgenerators to the power system; is the power output of the thgenerator. The ELD problem consists in minimizing subjectto the following constraints

1) Real Power Balance Constraint:

(2)

The transmission loss may be expressed using B-coefficientsas

(3)

2) Generator Capacity Constraints: The power generated byeach generator shall be within their lower limit and upperlimit . So that

(4)

B. ELDPOZRR

The objective function of this type of ELD problem issame as mentioned in ELDQCTL (1). Here the objective func-tion is to be minimized subject to the following constraints.

1) Real Power Balance Constraint: The real power balanceconstraint remains the same as in (2).

2) Generator Capacity Constraints: This constraint remainsunchanged as given in (4).

3) Ramp Rate Limit Constraints: The power generated, ,by the th generator in certain interval may not exceed that ofprevious interval by more than a certain amount , theup-ramp limit and neither may it be less than that of the previous

interval by more than some amount the down-ramp limitof the generator. These give rise to the following constraints.

As generation increases

(5)

As generation decreases

(6)

and

(7)

4) Prohibited Operating Zone: The prohibited operatingzones are the range of output power of a generator where theoperation causes undue vibration of the turbine shaft. Generallysuch vibration occurs at the point of opening or closing ofthe steam valve which might cause damage to the shaft andbearings. It is difficult to determine the exact prohibited zoneby actual testing or from operating records. Normally operationis avoided in such regions. Hence mathematically the feasibleoperating zones of unit can be described as follows:

(8)

where represents the number of prohibited operating zones ofunit is the upper limit of th prohibited oper-ating zone of th unit. is the lower limit of th prohibitedoperating zone of th unit. Total number of prohibited operatingzone of th unit is .

C. ELDVPL

In ELD with “valve point loadings”, objective function isrepresented by a more complex formula, given as in (9) atthe bottom of the page. Variation of fuel cost “ ” due tovalve point loading with the change of generation value isshown in Fig. 1. The objective of ELDVPL is to minimize of(9) subject to the same set of constraints given in (2) and (4) asin ELDQCTL.

D. ELDVPLMF

For a power system with generators and fuel optionsfor each unit, the cost function of the generator with valve-point

(9)


Fig. 1. Input-output curve with valve-point loading. a, b, c, d, e—valve points.

loading is expressed as (10) at the bottom of the page, whereand are the minimum and maximum power genera-

tion limits of the th generator with fuel option , respectively;, , , , and are the fuel-cost coefficients of gener-

ator for fuel .The above objective function is to be minimized subject to

the same constraints as mentioned in (2)–(4).

E. Calculation for Slack Generator

Let N committed generating units deliver the power outputsubject to their respective energy balance constraints (2) and thecapacity constraints (4). Assuming the power loadings of first

(N-1) generators as specified, the power level of generator(i.e., slack generator) is given by

(11)

The transmission loss is a function of all the generator out-puts including the dependent generator and it is given by

(12)

Expanding and rearranging, (11) becomes (13) at the bottom ofthe page. The loading of the dependent generator (i.e., ) canthen be found by solving (13) using standard algebraic method.

Above equation can be simplified as

(14)

where we see the last equation at the bottom of the page. Thepositive roots of the equation are obtained as

where (15)

(10)

(13)


To satisfy the equality constraint (11), the positive root of (15)is chosen as output of the th generator.

III. BIOGEOGRAPHY

Biogeography describes how species migrate from one islandto another, how new species arise, and how species become ex-tinct. A habitat is any island (area) that is geographically iso-lated from other islands. Areas that are well suited as residencesfor biological species are said to have a high habitat suitabilityindex (HSI). Factors that influence HSI include rainfall, diversityof vegetation, topographic features, land area, and temperature.The variables that characterize habitability are called suitabilityindex variables (SIVs). SIVs can be considered as the indepen-dent variables of the habitat, and HSI can be calculated usingthese variables. Habitats with a high HSI tend to have a largenumber of species, while those with a low HSI have a smallnumber of species. Habitats with a high HSI have many speciesthat migrate to nearby habitats, simply by virtue of the largenumber of species that they host. Migration of some speciesfrom one habitat to other habitat is known as emigration process.When some species enters into one habitat from any other out-side habitat, it is known as immigration process. Habitats witha high HSI have a low species immigration rate because theyare already nearly saturated with species. Therefore, high HSIhabitats are more static in their species distribution than low HSIhabitats. By the same token, high HSI habitats have a high em-igration rate. Habitats with a low HSI have a high species im-migration rate because of their sparse populations. This immi-gration of new species to low HSI habitats may raise the HSI ofthat habitat, because the suitability of a habitat is proportionalto its biological diversity. Here, Fig. 2 illustrates a model ofspecies abundance in a single habitat. Let us consider the im-migration graph of Fig. 2. The maximum possible immigrationrate to the habitat is I, which occurs when there are zero speciesin the habitat. If a habitat has less number of species, then muchlarger amount of species from other habitat can enter into thathabitat, so immigration rate is higher at that time. As the numberof species increases, the habitat becomes more crowded, andfewer species are able to successfully survive after immigrationto the habitat, and the immigration rate decreases. The largestpossible number of species that the habitat can support is ,at which point the immigration rate becomes zero, because nomore species can enter into that habitat after that species count.Now consider the emigration graph. If there are no species in thehabitat, then there is no species in that habitat that can shift toother habitat, so the emigration rate must be zero. As the numberof species increases, the habitat becomes more crowded, morespecies are able to leave the habitat to explore other possible res-idences, and the emigration rate increases. The maximum emi-

gration rate is E, which occurs when number of species is .The equilibrium number of species is , at which point the im-migration and emigration rates are equal. In Fig. 2 immigrationand emigration lines, graphically have been shown as straightlines but, in general, they might be more complicated curves.However, the simple model gives us a general description of theprocess of immigration and emigration. In BBO algorithm, cal-culation of emigration rate and immigration rate is important asthese play vital role to select habitats whose SIVs will undergomigration operation.

Mathematically the concept of emigration and immigrationcan be represented by a probabilistic model. Let us consider theprobability that the habitat contains exactly species at .

changes from time to time as follows:

(16)

where and are the immigration and emigration rates whenthere are species in the habitat. This equation holds becausein order to have species at time , one of the followingconditions must hold:

1) there were species at time , and no immigration or em-igration occurred between and ;

2) there were species at time , and one species im-migrated;

3) there were species at time , and one species emi-grated.

If time is small enough so that the probability of more thanone immigration or emigration can be ignored then taking thelimit of (16) as gives (17) at the bottom of the page.From the straight-line graph of Fig. 2, the equation for emigra-tion rate and immigration rate for number of speciescan be written as per the following way:

(18)

(19)

When value of , then combining (18) and (19)

(20)

IV. BIOGEOGRAPHY-BASED OPTIMIZATION (BBO)

This section describes development of biogeography-basedoptimization technique and the different steps involved therein.

(17)


Fig. 2. Species model of a single habitat.

Methodology of application of BBO technique to different typesof ELD problems has also been presented in this section.

BBO concept is based on the two major steps, e.g., migrationand mutation as discussed below.

A. Migration

In BBO algorithm a population of candidate solution can berepresented as vectors of real numbers. Each real number in thearray is considered as one (SIV). Using this SIV, the fitness ofeach set of candidate solution, i.e., HSI value can be evaluated.In an optimization problem high HSI solutions represent betterquality solution, and low HSI solutions represent an inferior so-lution.

The emigration and immigration rates of each solution areused to probabilistically share information between habitats.With probability , known as habitat modification prob-ability, each solution can be modified based on other solutions.According to BBO if a given solution is selected for modi-fication, then its immigration rate is used to probabilisticallydecide whether or not to modify each suitability index variable(SIV) in that solution. After selecting the SIV for modification,emigration rates of other solutions are used to select whichsolutions among the habitat set will migrate randomly chosenSIVs to the selected solution . In order to prevent the best so-lutions from being corrupted by immigration process, some kindof elitism is kept in BBO algorithm. Here, best habitat sets, i.e.,those habitats whose HSI are best, are kept as it is without mi-gration operation after each iteration. This operation is knownas elitism operation.

B. Mutation

It is well known that due to some natural calamities or otherevents HSI of natural habitat might get changed suddenly.In BBO such an event is represented by mutation of SIV andspecies count probabilities are used to determine mutationrates. The probabilities of each species count can be calculatedusing the differential equation of (17). Each habitat member hasan associated probability, which indicates the likelihood that itexists as a solution for a given problem. If the probability of agiven solution is very low, then that solution is likely to mutateto some other solution. Similarly if the probability of someother solution is high, then that solution has very little chanceto mutate. So it can be said that very high HSI solution and very

low HSI solutions have less chance to create more improvedSIV in the later stage. But medium HSI solutions have betterchance to create much better solutions after mutation operation.Mutation rate of each set of solution can be calculated in termsof species count probability using the following equation:

(21)

where is a user-defined parameter. This mutation schemetends to increase diversity among the habitats. Without thismodification, the highly probable solutions will tend to be moredominant in the total habitat. This mutation approach makesboth low and high HSI solutions likely to mutate, which givesa chance of improving both types of solutions in comparisonto their earlier value. Few kind of elitism is kept in mutationprocess to save the features of a solution, so if a solutionbecomes inferior after mutation process, then previous solution(solution of that set before mutation) can be reverted back tothat place again if needed. In ELD problem, if a solution is se-lected for mutation, then it is replaced by a randomly generatednew solution set. Other than this, any other mutation schemethat has been implemented for GAs can be implemented forBBO.

C. BBO Algorithm

The BBO algorithm can be described in the following way.Step 1) Initialize the BBO parameters like habitat modifica-

tion probability , mutation probability, max-imum mutation rate , max immigration rate

, max emigration rate , lower bound and upperbound for immigration probability per gene, stepsize for numerical integration , number of habitat

, number of SIV m, elitism parameter “p” whichindicates the number of best habitats to be retainedin the habitat matrix as it is, from one generation tothe next without performing migration operations onthem, etc. Set maximum number of iteration. Gen-erate the SIVs of the given problem within their fea-sible region using random number. A complete so-lution consisting of SIVs is known as one habitat .There are several numbers of habitats to search theoptimum result.

Step 2) Suppose we are minimizing a functionand . Initialize several num-

bers of habitats depending upon the habitat sizewithin feasible region. Each habitat represents apotential solution to the given problem. So totalhabitat in matrix form is written in the followingforms:

Step 3) Calculate the HSI value for each habitat of the pop-ulation set for given emigration rate , immigration


rate . For the function , HSI of allSIV sets is calculated as per the following way:

Calculate the number of valid species out of all habi-tats using their HSI values. Those habitats, whosefitness values, i.e., HSI values, are finite, are consid-ered as valid species .

Step 4) Based on the optimum HSI value, elite habitats areidentified.

Step 5) Probabilistically immigration rate and emigrationrate are used to modify each non-elite habitat usingmigration operation. The probability that a habitat

is modified is proportional to its immigration rateand the probability that the source of the modi-

fication comes from a habitat is proportional tothe emigration rate . Habitat modification usingmigration operation can be described as follows.

Select a habitat with probability proportional to

If is selected

For to

Select another habitat with probabilityproportional to

If is selected

Randomly select an SIV from habitat

Replace a random SIV in with that selectedSIV of

end

end

end

From this algorithm, we note that elitism can be im-plemented by setting for the best habitats.After each habitat is modified, its feasibility as aproblem solution should be verified. If it does notrepresent a feasible solution, then the above proce-dure is ignored and the same procedure is performedagain in order to map it to the set of feasible solu-tions. After modification of each non-elite habitatusing migration operation, each HSI is recomputed.

Step 6) For each habitat, the species count probability isupdated using (17). Mutation operation is per-formed on each non-elite habitat as discussed inSection IV-B and HSI value of each habitat is com-puted again. Mutation operation can be describedas follows.

For to

For to

Use and to compute the probability using(17)

Select a with probability proportional to

If is selected

Replace with a randomly generated SIVwithin its feasible region

end

end

end

As with habitat modification, elitism can be imple-mented by setting the probability of mutation selec-tion to zero for the best habitats. After eachhabitat is modified, its feasibility as a problem solu-tion should be verified. If it does not represent a fea-sible solution, then the above step is ignored and theabove-mentioned method is applied again in orderto map it to the set of feasible solutions.

Step 7) Go to step 3) for the next iteration. This loop can beterminated after a predefined number of iterationshave been found.

D. BBO Algorithm for ELD Problem

In this section, a new approach to implement the BBO al-gorithm will be described for solving the ELD problems. Es-pecially, a suggestion will be given on how to deal with theequality and inequality constraints of the ELD problems whenmodifying each individual’s search point in the BBO algorithm.The process of the BBO algorithm can be summarized as fol-lows.

1) Representation of the SIV: Since the decision variables forthe ELD problems are real power generations, they are used torepresent individual habitat. The real power output of all gen-erators is represented as the SIV in a habitat. For initialization,choose number of SIV of BBO algorithm , number of habitat

.The complete habitat set is represented in the form of the

following matrix:

i.e., see the equation at the bottom of the next page, whereand .

Here is the position vector of the habitat . Each habitat isone of the possible solutions for the ELD problem. Size of thehabitat is equivalent to the population size of GA. The element

of is the th position component of habitat or in otherwords is the th SIV of the th habitat. represents thereal power generation of generator of the th habitat set .

2) Initialization of the SIV: Each element of the Habitat ma-trix, i.e., each SIV of a given habitat set , is initialized ran-domly within the effective real power operating limits. The ini-tialization is based on (4) for generators without ramp rate limitsand based on (4), (7) for generators with ramp rate limits.

Now the steps of algorithm to solve ELD problems are givenbelow.


Step 1) For initialization, choose the number of generatorunits, i.e., number of SIV is , number of habitatis . Specify maximum and minimum capacity ofeach generator, power demand, B-coefficients ma-trix for calculation of transmission loss. Also ini-tialize the BBO parameters like habitat modifica-tion probability , mutation probability, max-imum mutation rate , max immigration rate

, max emigration rate , lower bound for immi-gration probability per gene, upper bound for immi-gration probability per gene, step size for numericalintegration , elitism parameter “p”, etc. Set max-imum number of iteration.

Step 2) Each SIV of a given habitat of matrix is initializedusing the concept mentioned in “Initialization of theSIV”. Each habitat set of matrix should satisfyequality constraint (2) using the concept of slackgenerator as mentioned in Section II-E. Each habitatrepresents a potential solution to the given problem.

Step 3) Calculate the HSI for each habitat set of the totalhabitat set for given emigration rate , immigrationrate . HSI represent the fuel cost of the generatorsin the power system for a particular power demand.Here, indicates the fuel cost due to the th setof generation value (i.e., th set of habitat matrix )in $/h. If there are units that must be operated toprovide power to loads, then the th individualcan be defined as follows:

The dimension of the habitat matrix is . Allthese components in each individual are representedas real values. The matrix represents the total habitatset.

Step 4) Based on the HSI (fuel cost in case of ELD problem),value elite habitats are identified. Here elite term isused to indicate those habitat sets of generator poweroutput, which give best fuel cost. Top “p” habitatsets are kept as it is after individual iteration withoutmaking any modification on it. Identification of validspecies is a little interesting. Those habitats, whosefitness values, i.e., HSI values, are finite, are consid-ered as valid species in ELD problem.

Step 5) Probabilistically perform migration operation onthose SIVs of each non-elite habitats, selected formigration. How to select any SIV for migrationoperation is described below.1) First select lower and upper value of immigra-

tion rate and , respectively:-Next calculate Species count:-

for to

if fuel cost of habitat set i,

SpeciesCount of habitat ;

else

SpeciesCount of habitat ;

end

end

2) Then calculate value of and for each habitatset:-

for to

(1-Species Count of habitat i/Habitat size );

Species Count of habitat i/Habitat size ;

end

3) Next calculate from which habitat and whichSIV to be selected for newly generated habitatafter migration:-

for to

if a randomly generated number is less than habitatmodification probability , then followingoperations are calculated

Normalize the immigration rate using the followingformula:-.


Pick up a habitat from which to obtain a feature:-

for to

if a randomly generated

;

;

;

while and

;

(SelectIndex);

end

;

else

;

end

end

end

After migration operation, new habitat set isgenerated. In ELD problems, these representnew modified generation values of generators

.4) Operating Limit Constraint Is Satisfied in the

Following Manner:-

If output of th generator, maximum capacity of thgenerator,

end

If output of th generator minimum capacity of thgenerator

end

If generation value of th generator is within itsmaximum and minimum generation capacity

end

Equality constraint (2) is satisfied using conceptof slack generator as mentioned in Section II-E.

Step 6) Species count probability of each habitat is updatedusing (17). Mutation operation is performed on thenon-elite habitat. If mutation rate as calculated using(21) of any habitat is greater than a randomly gener-ated number, then that habitat is selected for muta-tion. In mutation operation, that habitat set which is

selected for mutation is simply replaced by anotherrandomly generated new habitat set that satisfiesboth equality constraint and inequality constraints ofELD problems. HSI value of each new habitat set isrecomputed, i.e., fuel cost of each power generationset.

Step 7) Go to step 3) for the next iteration. This loop can beterminated after a predefined number of iterations.

After each habitat is modified (steps 5 and 6), its feasibility asa problem solution should be verified, i.e., each SIV should sat-isfy different operational constraints of generator as mentionedin the specific problem. Equality constraints should also be sat-isfied.

V. NUMERICAL EXAMPLE AND SIMULATION RESULT

Proposed BBO algorithm has been applied to ELD problemsin four different test cases for verifying its feasibility. Theseare a 20-generator system, a six-generator system, a 40-gener-ator system, and a ten-generator system. Here, the result ob-tained from proposed BBO method has been compared withSOH_PSO [12] and other methods for the 40-generator systems;with Hopfield model [4] and other techniques for the 20-gener-ator systems; with NPSO_LRS [13] and other techniques for theten-generator systems and with SOH_PSO [12] and other pre-viously developed techniques for the six-generator systems. Areasonable B-loss coefficients matrix of power system networkhas been employed to calculate the transmission loss. The soft-ware has been written in MATLAB-7 language and executed ona 2.3-GHz Pentium IV personal computer with 512-MB RAM.

A. Description of the Test Systems

1) Test Case 1: A system with six generators with ramprate limit and prohibited operating zone is used here. The inputdata have been adopted from [15]. The load demand is 1263MW. Results obtained from proposed BBO, PSO [15] and newcoding-based modified PSO [14] and other methods have beenpresented here. Their best solutions are shown in Table I. Theconvergence characteristic of the six-generator systems in caseof BBO algorithm is shown in Fig. 3.

2) Test Case 2: A system with 40 generators with valve pointloading is used here. The input data are given in [16]. The loaddemand is 10 500 MW. Transmission loss has not been con-sidered here. The result obtained from proposed BBO methodhas been compared with NPSO-LRS [13], SOH_PSO [12], andother methods. Their best solutions are shown in Table III. Aconvergence characteristic of the 40-generator systems in caseof BBO algorithm is shown in Fig. 4.

3) Test Case 3: A simple system with ten thermal units isconsidered here. The input data are taken from [17]. The loaddemand is 2700 MW. Transmission loss has not been consid-ered here. The result obtained from the proposed BBO, differentPSO techniques [13], and different GA [17] methods are shownin Table IX. Convergence characteristic of the 20-generator sys-tems in case of BBO algorithm is shown in Fig. 5.

4) Test Case 4: A system with 20 generators is taken here.The input data of the system are given in [4]. The load demand


TABLE IBEST POWER OUTPUT FOR SIX-GENERATOR SYSTEM ��

Fig. 3. Convergence characteristic of six-generator system.

is 2500 MW. The result obtained from proposed BBO algo-rithm has been compared with Lambda iteration [4] and Hop-field model [4]. Their best solutions are shown in Table XII. Aconvergence characteristic of the 20-generator systems in caseof BBO algorithm is shown in Fig. 6.

B. Determination of Parameters for BBO Algorithm

To get optimal solution using the BBO algorithm, the bestvalues of the parameters like mutation probability, step of inte-gration , and habitat size have to be determined. To findoptimum values for “step size of integration ” and “mutationprobability”, the following procedures have been applied.

1) The habitat size is fixed at 50.

Fig. 4. Convergence characteristic of 40-generator system.

Fig. 5. Convergence characteristic of ten-generator system.

Fig. 6. Convergence characteristic of 20-generator system.

2) Step of integration is increased from 0.1 to 2 in suitablesteps as shown in Tables V and VI and mutation probabilityis changed to three different values of 0.5, 0.05, and 0.005.Performance of BBO algorithm in ELDVPL system iscalculated for all the above-mentioned combinations. For


TABLE IIIBEST POWER OUTPUT FOR 40-GENERATOR SYSTEM ��

each combination, 50 independent trials have been madewith 1000 iterations per trial.

TABLE IXBEST POWER OUTPUT FOR TEN-GENERATOR SYSTEM ��

TABLE XIIBEST POWER OUTPUT FOR 20-GENERATOR SYSTEM ��

3) Step of integration is again increased from 0.1 to 2 in samesteps and the same problem as mentioned above and mu-tation probability is not considered in that case.

The minimum generation costs for this system reported so farare 121 501.14 $/h [12], 121 664.4308 $/h [13], and 121 704.7391 $/h [13]. In case of BBO algorithm, based on the simu-lation results obtained for different combination of parameters


TABLE VINFLUENCE OF PARAMETERS ON BBO PERFORMANCE

TABLE VIINFLUENCE OF PARAMETERS ON BBO PERFORMANCE

given in Tables V and VI, step size of integration 1 and muta-tion probability 0.005 gives optimal generation cost more con-sistently. The obtained minimum generation cost 121 426.953$/h is also less among the remaining cases.

C. Effect of Habitat Size on BBO Algorithm

Change in habitat size affects the performance of the BBO al-gorithm. Too large or a small habitat size may not be capable of

TABLE VIIEFFECT OF HABITAT SIZE ON RESULTS OF 40-GENERATOR SYSTEM

TABLE XCOMPARISON AMONG DIFFERENT METHODS

AFTER 100 TRIALS (TEN-GENERATOR SYSTEM)

searching for the minimum, particularly in complex multimodalproblems. The optimum habitat size is found to be related to theproblem dimension and complexity.

Table VII shows the performance of the BBO algorithm fordifferent habitat sizes. Tests were carried out 50 times each fora habitat size of 20, 50, 100, and 200 for the 40-unit system.

A habitat size of 50 resulted in achieving global solutionsmore consistently for the test system. Increasing the habitat sizebeyond this value did not produce any significant improvement;rather, it increases the simulation time which is not desirablein real-time problems. Hence, after a number of careful exper-imentation, the following optimum values of BBO parametershave finally been settled.

Habitat size , Habitat Modification Probability, Immigration Probability bounds per gene

, step size for numerical integration ,maximum immigration emigration rate for each island ,and Mutation Probability .

D. Comparative Study

1) Solution Quality: From Tables I, III, and IX it is clearthat the minimum cost achieved by BBO are 15 443.0963 $/h,121 426.953 $/h, and 605.6387 $/h for Test Case 1, 2, and 3,respectively. Those are best and less than reported in recentliterature [12]–[14], [17]. In Test Case 4, solution obtained byBBO is 15 456.7926 $/h which is very close to results obtainedby Lambda iteration and Hopfield method. Tables II, IV, andX also show that the average costs produced by BBO are leastcompared with other methods, emphasizing its better solutionquality. In Test Case 1, 2, 3, and 4, average costs obtained byBBO are 15 443.0964 $/h, 121 508.0325 $/h, 605.8622 $/h, and62 456.7928 $/h, respectively. These are less than results ob-tained by other methods. Table XIII shows that the performanceof BBO method is consistent for large convex type system also.As the average cost of generation in BBO is better for bothconvex and non-convex ELD problems, it only indicates its


TABLE XIIISTATISTICS FOR 20-GENERATOR SYSTEM

TABLE IICOMPARISON AMONG DIFFERENT METHODS

AFTER 50 TRIALS (SIX-GENERATOR SYSTEM)

TABLE IVCOMPARISON AMONG DIFFERENT METHODS

AFTER 50 TRIALS (40-GENERATOR SYSTEM)

ability to reach global minima in a consistent manner and itsbetter convergence characteristic.

2) Computational Efficiency: Tables I, III, and IX showthat the minimum cost achieved by BBO are 15 443.0963 $/h,121 426.953 $/h, and 605.6387 $/h for Test Case 1, 2, and 3,respectively. Those are less as compared to the reported resultsin recent literature. Table XII shows that the minimum costachieved by the BBO algorithm is 62 456.7926 $/h in Test Case4 which is not best, but very closer to previously mentionedmethods. Again power mismatches between total power gen-eration and is least in BBOcompared to other described methods. The BBO approach isalso efficient as far as computational time is concerned. Timerequirement is quite less and either comparable or better thanother mentioned methods. So as a whole, it can be said thatthe BBO method is computationally efficient than previouslymentioned methods.

3) Robustness: Since initialization of habitat is performedusing random numbers in case of stochastic simulation tech-niques, so randomness is an inherent property of these tech-niques. Hence the performances of stochastic search algorithmsare judged out of a number of trials. Many trials with differentinitial habitats have been carried out to test the consistency ofthe BBO algorithm. Tables VIII and XI show the frequency ofattaining cost within different ranges for Test Case 2 and 3, outof 50 and 100 independent trials, respectively. It can be seen thatBBO approach is robust as it reaches to minimum cost 38 and100 times out of 50 and 100 times, respectively.

From the simulation result of BBO algorithm in solving ELDproblems, we cannot conclude that BBO is universally better

TABLE VIIIFREQUENCY OF CONVERGENCE, 40-GENERATOR SYSTEM OUT OF 50 TRIALS

TABLE XIFREQUENCY OF CONVERGENCE FOR TEN-GENERATOR

SYSTEM OUT OF 100 TRIALS

than other methods. It has been seen that performance of BBOalgorithm in complex multi-modal, non-convex type problemis much better than other optimization techniques mentioned inthis paper. But its performance is not very effective than othermethods in simple system with quadratic cost function (TestCase 4). In Test Case 4 its performance is not better than theother mentioned algorithms in terms of fuel cost, but it mini-mizes misfitness between power generation and power demand.So it may be said that performance of BBO is not much supe-rior than other mentioned methods in case of simple cost func-tions as it is in case of non-convex, complex type cost functions.However, it may be possible in future work to quantify the per-formance of BBO relative to other algorithms for problems withthis type of specific features. The good performance of BBO onthe ELD problems provides some evidence that BBO theory canbe successfully applied to various practical power system opti-mization problems in the future.

VI. CONCLUSION

The BBO method has been successfully implemented to solvedifferent convex and non-convex ELD problems with the gener-ator constraints. The BBO algorithm has the ability to find thebetter quality solution and has better convergence characteris-tics, computational efficiency, and robustness. Many nonlinearcharacteristics of the generator such as ramp rate limits, valvepoint loadings, multi-fuel options, prohibited operating zone,etc. have been considered. It is clear from the results obtained


by different trials that the proposed BBO method has good con-vergence property and can avoid the shortcoming of prematureconvergence of other optimization techniques to obtain betterquality solution. Due to these properties, the BBO method inthe future can be tried for solution of complex unit commitment,dynamic ELD problems in the search of better quality results.

ACKNOWLEDGMENT

The author would like to thank D. Simon, Associate Professorin the Electrical and Computer Engineering Department, Cleve-land State University, whose paper on BBO has helped a lot tocarry out this work.

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Aniruddha Bhattacharya (M’09) received theB.Sc. Engg. degree in electrical engineering fromthe Regional Institute of Technology, Jamshedpur,India, in 2000 and the M.E.E. degree in electricalpower system from Jadavpur University, Kolkata,India, in 2008. He is currently pursuing the Ph.D.degree in the Department of Electrical Engineeringat Jadavpur University.

His employment experience includes the SiemensMetering Limited, India; Jindal Steel & PowerLimited, Raigarh, India; Bankura Unnyani Institute

of Engineering, Bankura, India; and Dr. B. C. Roy Engineering College,Durgapur, India. His areas of interest include power system load flow, optimalpower flow, economic load dispatch, and soft computing applications todifferent power system problems.

Pranab Kumar Chattopadhyay received the M.E.Edegree in electrical power system from Jadavpur Uni-versity, Kolkata, India, in 1971.

He is currently working as a Professor in theDepartment of Electrical Engineering, JadavpurUniversity. His areas of interest include applicationof soft computing techniques to different powersystem problems.

biogeography-based optimization for different economic load dispatch problems

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